Question

simplify.
\\((2w^{-6}x^{5})^{-5}\\)
write your answer using only positive exponents.

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, we can apply this to \((2w^{-6}x^{5})^{-5}\) as follows:
\((2w^{-6}x^{5})^{-5}=2^{-5}(w^{-6})^{-5}(x^{5})^{-5}\)

Step2: Simplify each term using power of a power rule

The power of a power rule is \((a^m)^n=a^{mn}\).

• For \(2^{-5}\), we know that \(a^{-n}=\frac{1}{a^{n}}\), so \(2^{-5}=\frac{1}{2^{5}}=\frac{1}{32}\) (but we'll keep it as \(2^{-5}\) for now and combine later).
• For \((w^{-6})^{-5}\), using the power of a power rule: \((w^{-6})^{-5}=w^{(-6)\times(-5)} = w^{30}\)
• For \((x^{5})^{-5}\), using the power of a power rule: \((x^{5})^{-5}=x^{5\times(-5)}=x^{-25}\)

So now we have \(2^{-5}w^{30}x^{-25}\)

Step3: Convert negative exponents to positive

We know that \(a^{-n}=\frac{1}{a^{n}}\), so \(2^{-5}=\frac{1}{2^{5}}=\frac{1}{32}\) and \(x^{-25}=\frac{1}{x^{25}}\). But we can also rewrite the expression with positive exponents by moving the terms with negative exponents to the denominator or numerator as needed.

Combining the terms:
\(2^{-5}w^{30}x^{-25}=\frac{w^{30}}{2^{5}x^{25}}\)

Since \(2^{5} = 32\), we get \(\frac{w^{30}}{32x^{25}}\)

Answer:

\(\frac{w^{30}}{32x^{25}}\)